Preparation

Scrape seeds from vanilla bean with tip of a paring knife into a small heavy saucepan. Add pod, milk, and 3 Tbsp sugar and bring just to a boil, whisking to dissolve sugar.

Meanwhile, whisk together flour, cornstarch, remaining sugar, and a pinch of salt in a bowl. Whisk together whole egg and yolk in another bowl, then whisk in flour mixture until smooth.

Add one third of hot milk mixture to yolk mixture (to temper) in
a slow stream, whisking, then add to milk mixture remaining
in saucepan, whisking. Bring custard to a boil over medium heat, whisking, then boil, whisking, 2 minutes. Remove from heat and stir in butter until melted.

Transfer pastry cream to a bowl and chill, its surface covered with wax paper, until no longer hot, at least 30 minutes.

Make tarts:

Preheat oven to 400°F with racks in upper and lower thirds. Line 2 large baking sheets with parchment paper.

Peel and core apples, then cut into 1/8-inch-thick wedges.

Roll out pastry on a lightly floured surface with a lightly floured rolling pin 1/8 inch thick and cut out 8 (6-inch) rounds using inverted bowl or plate as a guide. Punch out center of each round with cookie cutter and transfer to baking sheets.

Discard vanilla pod and transfer pastry cream to pastry bag.
Snip a 1/3-inch opening and pipe pastry cream onto pastry in a ring around center (there will be some cream left over). Arrange overlapping apple slices on pastry, covering the cream, then brush with melted butter.

Bake tarts, switching position of sheets halfway through, until edges of apples are pale golden, about 15 minutes.

Meanwhile, stir together corn syrup and water until smooth.

Remove baking sheets from oven and flip each tart over with a large spatula. Generously brush pastry with some of corn syrup mixture.

Continue to bake, switching position of sheets halfway through, until pastry is shiny and golden, about 15 minutes more. Cool
10 minutes.

Serve tarts, pastry side up, warm or at room temperature.

Cooks' notes:
Pastry cream can be chilled up to 3 days. Stir before using.
Pastry can be rolled out and cut 1 week ahead and frozen, tightly wrapped in plastic wrap. Do not thaw.
Tarts can be baked 4 hours ahead. Reheat in a 350°F oven 5 to 10 minutes.